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WASHINGTON (CNN) -- The top Republicans in both the House and Senate are indicating they don't support language in an immigration bill that would make entering the country illegally a felony.

The proposal has drawn the ire of pro-immigrant groups that have staged a wave of protests in recent weeks.

The provision making illegal immigration a felony was contained in an immigration reform bill passed by the House in December. But in a joint statement issued Tuesday evening, House Speaker Dennis Hastert of Illinois and Senate Majority Leader Bill Frist of Tennessee both indicated they wanted the language dropped.

Frist and Hastert also criticized House Democrats, who, they said, opposed efforts by Republicans to strip the provision from the bill before it passed.

Frist and Hastert did not specify whether they wanted unlawful presence in the United States to be a misdemeanor or carry a lesser penalty.

Their statement was also silent on the question of whether they had come to any agreement on two issues that have split Republicans -- creating a guest-worker program, or allowing undocumented immigrants in the country illegally to work their way toward legal status.

The provision making illegal immigration a felony was part of a bill pushed by House Judiciary Committee Chairman James Sensenbrenner, a Wisconsin Republican. It passed the House in December by a vote of 239-182, with only 36 Democrats supporting the final version of the measure.

Responding to Tuesday's criticism of Democrats by Hastert and Frist, Jennifer Crider, a spokeswoman for House Minority Leader Nancy Pelosi of California, said "no amount of spin can change the fact that Republicans wrote and passed the Sensenbrenner bill, which criminalizes an entire population."

Crider also said Republicans "are feeling the heat" after demonstrations that brought out hundreds of thousands of protesters Monday at rallies in at least 140 cities in more than 39 states. (Full story)

Sensenbrenner, who sponsored the provision making illegal immigration a felony, said last week that he tried to remove it from the bill in December and remains open to making the change as the House and Senate try to reach an agreement on a final bill.

Republican Rep. Tom Tancredo of Colorado, a leading advocate of cracking down on illegal immigration, has accused Democrats of trying to keep the felony provision in the bill as a "poison pill."

But Sen. Edward Kennedy on Tuesday dismissed such characterizations.

"Actions speak louder than words, and there's no running away from the fact that the Republican House passed a bill and Senator Frist offered one that criminalizes immigrants," the Massachusetts Democrat said.

"This debate shouldn't be about making criminals out of hard-working families ... but rather about strengthening our national security and enacting a law that reflects our best values and our humanity," he said in a written statement.

Sensenbrenner's bill also calls for building 700 miles of security fence along the Mexican border and would also make assisting illegal immigrants a felony.

It does not include a guest-worker provision, as President Bush has called for, or a legalization process for people already in the United States illegally. Critics dismiss that idea as "amnesty," while supporters call it "earned citizenship." (Watch how illegal labor impacts the economy -- 2:42)

House GOP aides said Tuesday that language aimed at punishing people who help illegal immigrants was aimed at smugglers who bring people across the border, not at charities who assist the migrants.

As protests against the House bill mounted in late March, the White House and the Republican National Committee raised concerns that the anti-immigration sentiment coming from some corners of the GOP would turn off Latino voters that Bush and his political team have worked hard to court.

Senate attempts to pass an immigration reform bill stalled last week when a measure establishing a guest-worker program and a mechanism for legalization failed to overcome opposition from conservative Republicans.